1. Volume using disk integration

Hello,
I have to find the volume of the area between x=1 and y=(e^x)+1 rotated around both the x and y axis. The x axis worked fine, and I got about 10.83pi. However when I try to rotate the area around the y axis, I get the area between x=1 and x=ln(y-1) for 0<y<1, and when I attempt to integrate I get a large integral with ln(y-1) in it, and after subbing in 0 I get a negative number and therefore an error. What am I doing wrong? All help is appreciated

Hello,
I have to find the volume of the area between x=1 and y=(e^x)+1 rotated around both the x and y axis. The x axis worked fine, and I got about 10.83pi. However when I try to rotate the area around the y axis, I get the area between x=1 and x=ln(y-1) for 0<y<1, and when I attempt to integrate I get a large integral with ln(y-1) in it, and after subbing in 0 I get a negative number and therefore an error. What am I doing wrong?
disks and washers about the y-axis ...

$\displaystyle \displaystyle V = \pi \int_0^2 1^2 \, dy + \pi \int_2^{e+1} 1^2 - [\ln(y-1)]^2 \, dy$

to check, method of cylindrical shells about the y-axis ...

$\displaystyle \displaystyle V = 2\pi \int_0^1 x(e^x + 1) \, dx$

3. one more way ...

big cylinder - "scooped" out volume

$\displaystyle \displaystyle V = \pi \int_0^{e+1} 1^2 \, dy - \pi \int_2^{e+1} [\ln(y-1)]^2 \, dy$